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Theorem sbalv 2167
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbalv.1 ([𝑦 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbalv ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem sbalv
StepHypRef Expression
1 sbal 2166 . 2 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2 sbalv.1 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
32albii 1820 . 2 (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
41, 3bitri 277 1 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wal 1535  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-11 2161
This theorem depends on definitions:  df-bi 209  df-ex 1781  df-sb 2070
This theorem is referenced by:  sbex  2288  sbmo  2697  sbabel  3005  mo5f  30238
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